We use a numerical optimization method to determine the control parameters
frequency, power, and coil position for the radio frequency (RF) induction
heating of the growth apparatus during sublimation growth of SiC single crystals
via physical vapor transport (PVT) (also called the modified Lely method). The
control parameters are determined to minimize a functional, tuning the radial
temperature gradient on the single crystal surface as well as the vertical temperature
gradient between SiC source and seed, both being crucial for high-quality
growth. The optimization is subject to constraints with respect to a required
temperature difference between source and seed, a required temperature range at
the seed, and an upper bound for the temperature in the entire apparatus. The
numerical computations use a stationary mathematical model for the heat transport,
including heat conduction, radiation, and RF heating to solve the forward
problem, and a Nelder-Mead method for optimization. A minimal radial temperature
gradient is found to coincide with a minimal temperature at the single
crystal surface, and a maximal temperature gradient between source and seed is
found to coincide with a low coil position.
This paper addresses the regularization of pointwise state constraints in optimal
control problems. By analyzing the associated dual problem, it is shown that the regularized problems
admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of
the regularized problem converges to the one of the original state constrained problem. The results
of our analysis are confirmed by numerical tests.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.
This paper is concerned with the state-constrained optimal control of the
two-dimensional thermistor problem, a quasi-linear coupled system
of a parabolic and elliptic PDE with mixed boundary conditions.
This system models the heating of a conducting material by means of direct current.
Existence, uniqueness and continuity for the state system are derived by employing
maximal elliptic and parabolic regularity. By similar arguments the
linearized state system is discussed, while the adjoint system involving measures
is investigated using a duality argument. These results allow to derive
first-order necessary conditions for the optimal control problem.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise linear functions. The state and the
adjoint state are discretized by linear finite elements. Approximation of order h in the L1-norm is proved in the
main result.
We consider a control constrained optimal control problem governed by a semilinear
elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of
diffuse-gray conductive-radiative heat transfer. The problem arises from the aim to optimize the
temperature gradient within crystal growth by the physical vapor transport (PVT) method. Based
on a minimum principle for the semilinear equation as well as L1-estimates for the weak solution,
we establish the existence of an optimal solution as well as necessary optimality conditions. The
theoretical results are illustrated by results of numerical computations.
We consider a control constrained optimal control problem governed by a semilinear
elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of
diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, secondorder
sufficient conditions are derived that account for strongly active sets. These conditions ensure
local optimality in a Ls-neighborhood whereby the underlying analysis allows to use weaker norms
than L?.
A linear-quadratic elliptic control problem with pointwise box constraints on the
state is considered. The state-constraints are treated by a Lavrentiev type regularization. It is
shown that the Lagrange multiplier associated with the regularized state-constraints are functions
in L2. Moreover, the convergence of the regularized controls is proven for regularization parameter
tending to zero. To solve the problem numerically, an interior point method and a primal-dual active
set strategy are implemented and treated in function space.
We consider a control- and state-constrained optimal control problem
governed by a semilinear
elliptic equation with nonlocal interface conditions.
These conditions occur during the
modeling of diffuse-gray conductive-radiative heat transfer.
The nonlocal radiation interface condition and the pointwise state-constraints
represent the particular features of this problem. To deal with the
state-constraints, continuity of the state is shown which allows to
derive first-order necessary conditions. Afterwards, we establish second-order
sufficient conditions that account for strongly active sets and
ensure local optimality in an $L^2$-neighborhood.
A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.
A state-constrained optimal control problem arising in the context of sublimation crystal growth is considered. The presence of pointwise state-constraints and nonlocal radiation interface conditions
constitutes the major issue of this problem. A regularity result of the state is presented that allows to
derive the optimality condition.
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.